You CAN take the first airport bus from Plaza Galicia at 06:00. BUT, CONSIDER that the trip usually takes about 40 - 45 minutes. The bus starts at Plaza Galicia, then stops at: the train station, bus station, and maybe three other in-city stops before heading out to the airport.
Thus, IMHO the taxi is the best way to get to the airport at, say 05:00 or 05:30, before bus service starts. This is especially the case if you are on that first 06:40 flight. BUT, you need to call the night before to reserve, to ensure someone is on duty.
Another option is to stay your final night at Hotel Garcas. It is about one kilometer from the airport terminal building. You can stay there on your last night, then walk to the airport as early as you choose. It winds up costing more than if you took the taxi, but might represent an economy if you were paying for a hotel in Santiago anyway.
This hotel is clean, convenient and well priced. Itis popular with folks leaving early in the morning.
When walking in on the CF, I usually stay there my final night before arriving in Santiago. It is just a few hundred meters off the Camino route, where it crosses the N-634 in the village of Lavacolla and descends into the forest. When you get the the road, just after the church, turn left and walk the shoulder / verge of the road. Hotel Garcas is the terracotta colored building on the right. You cannot miss it.
If arriving at Hotel Garcas to spend your last night before flying out, the easiest thing to do is to take the airport bus from Plaza de Galicia. Strangely, the FINAL AIRPORT BUS STOP, before the airport terminal is immediately in front of Hotel Garcas. You can get off the airport bus there to access the hotel.
Hope this helps.
